\( \left. \begin{array} { l } { \frac { \operatorname { Sin } ( 360 - 2 x ) \cdot \sin ( x ) } { \sin ( 90 + x ) } + 2 \cos ^ { 2 } ( 180 + x ) } \\ { = \ln 2 x ( - \sin x ) + 2 \cos ^ { 2 } x } \end{array} \right. \)

It seems that the input expression is not being recognized correctly due to its complexity. Let's break down the problem step by step and simplify the expressions involved. ### Given Equation We have the equation: \[ \frac{\sin(360 - 2x) \cdot \sin(x)}{\sin(90 + x)} + 2 \cos^2(180 + x) = \ln(2x)(-\sin(x)) + 2 \cos^2(x) \] ### Step 1: Simplify the Trigonometric Functions 1. **Simplify \(\sin(360 - 2x)\)**: \[ \sin(360 - 2x) = -\sin(2x) \] 2. **Simplify \(\sin(90 + x)\)**: \[ \sin(90 + x) = \cos(x) \] 3. **Simplify \(\cos(180 + x)\)**: \[ \cos(180 + x) = -\cos(x) \] Therefore, \[ \cos^2(180 + x) = \cos^2(x) \] ### Step 2: Substitute Back into the Equation Substituting these simplifications into the equation gives: \[ \frac{-\sin(2x) \cdot \sin(x)}{\cos(x)} + 2 \cos^2(x) = \ln(2x)(-\sin(x)) + 2 \cos^2(x) \] ### Step 3: Cancel Common Terms Notice that \(2 \cos^2(x)\) appears on both sides of the equation, so we can cancel it out: \[ \frac{-\sin(2x) \cdot \sin(x)}{\cos(x)} = \ln(2x)(-\sin(x)) \] ### Step 4: Further Simplification Assuming \(\sin(x) \neq 0\), we can divide both sides by \(-\sin(x)\): \[ \frac{\sin(2x)}{\cos(x)} = \ln(2x) \] ### Step 5: Solve the Equation Now we need to solve the equation: \[ \sin(2x) = \ln(2x) \cos(x) \] This equation is transcendental and may not have a straightforward algebraic solution. We can analyze it graphically or numerically to find the values of \(x\). ### Conclusion To find the solutions for \(x\), we can use numerical methods or graphing techniques. If you would like to proceed with a numerical solution or further analysis, please let me know!